\(\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx\) [2289]

Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \]

[Out]

-2761/1920*(1-2*x)^(3/2)*(3+5*x)^(3/2)-251/800*(1-2*x)^(3/2)*(3+5*x)^(5/2)-3/50*(1-2*x)^(3/2)*(3+5*x)^(7/2)+36
74891/512000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-30371/5120*(1-2*x)^(3/2)*(3+5*x)^(1/2)+334081/51200*
(1-2*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222} \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {3674891 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}}-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac {30371 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120}+\frac {334081 \sqrt {1-2 x} \sqrt {5 x+3}}{51200} \]

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(5/2),x]

[Out]

(334081*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200 - (30371*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5120 - (2761*(1 - 2*x)^(3/2
)*(3 + 5*x)^(3/2))/1920 - (251*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/50 +
(3674891*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200*Sqrt[10])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps \begin{align*} \text {integral}& = -\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {251}{100} \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx \\ & = -\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {2761}{320} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx \\ & = -\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {30371 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{1280} \\ & = -\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {334081 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{10240} \\ & = \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400} \\ & = \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200 \sqrt {5}} \\ & = \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-3762261-4115415 x+16522420 x^2+37765600 x^3+33936000 x^4+11520000 x^5\right )-11024673 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1536000 \sqrt {3+5 x}} \]

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(5/2),x]

[Out]

(10*Sqrt[1 - 2*x]*(-3762261 - 4115415*x + 16522420*x^2 + 37765600*x^3 + 33936000*x^4 + 11520000*x^5) - 1102467
3*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(1536000*Sqrt[3 + 5*x])

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\left (2304000 x^{4}+5404800 x^{3}+4310240 x^{2}+718340 x -1254087\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{153600 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3674891 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (46080000 x^{4} \sqrt {-10 x^{2}-x +3}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-25081740 \sqrt {-10 x^{2}-x +3}\right )}{3072000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

[In]

int((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/153600*(2304000*x^4+5404800*x^3+4310240*x^2+718340*x-1254087)*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1
/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+3674891/1024000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2
)/(1-2*x)^(1/2)/(3+5*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{153600} \, {\left (2304000 \, x^{4} + 5404800 \, x^{3} + 4310240 \, x^{2} + 718340 \, x - 1254087\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3674891}{1024000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

[In]

integrate((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/153600*(2304000*x^4 + 5404800*x^3 + 4310240*x^2 + 718340*x - 1254087)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3674891
/1024000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

Sympy [F]

\[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((2+3*x)*(3+5*x)**(5/2)*(1-2*x)**(1/2),x)

[Out]

Integral(sqrt(1 - 2*x)*(3*x + 2)*(5*x + 3)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {3}{2} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {539}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1121}{384} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30371}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3674891}{1024000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30371}{51200} \, \sqrt {-10 \, x^{2} - x + 3} \]

[In]

integrate((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-3/2*(-10*x^2 - x + 3)^(3/2)*x^2 - 539/160*(-10*x^2 - x + 3)^(3/2)*x - 1121/384*(-10*x^2 - x + 3)^(3/2) + 3037
1/2560*sqrt(-10*x^2 - x + 3)*x - 3674891/1024000*sqrt(10)*arcsin(-20/11*x - 1/11) + 30371/51200*sqrt(-10*x^2 -
x + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).

Time = 0.35 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{2560000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {37}{384000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {351}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

[In]

integrate((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

1/2560000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5
*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 37/384000*sqrt(5)*(2*(4*(8*(
60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3))) + 57/8000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4
785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 351/2000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 27/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2} \,d x \]

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)*(5*x + 3)^(5/2),x)

[Out]

int((1 - 2*x)^(1/2)*(3*x + 2)*(5*x + 3)^(5/2), x)